The Capacity Region of the $L$ -User Gaussian Inverse Compute-and-Forward Problem

We consider an L-user multiple access channel where transmitter m has access to the linear equation u_m = ⊕_l=1^L f_mlw_l of independent messages w_l ∈ F_p^kl with f_ml ∈ F_p, and the destination wishes to recover all L messages. This problem may be motivated as the last hop in a network where relay nodes employ the compute-and-forward strategy and decode linear equations of messages; we seek to do the reverse and extract messages from sums over a multiple access channel. In particular, we exploit the particular form of dependencies between the equations at the different relays to improve the reliable communication rates beyond those achievable by simply forwarding all equations to the destination independently. The presented achievable rate region for the discrete memoryless channel model is shown to be capacity for the additive white Gaussian noise channel.